California Patty Melts

In a medium-size saute pan or skillet, melt the butter over medium heat.
Add the onion, cover, reduce the heat to low, and cook, stirring occasionally, for 15 minutes.
Remove cover, and continue to cook the onion over low heat until very soft and golden, about 45 minutes longer, stirring to keep onions from sticking to pan.
Start up grill or preheat a broiler.
While onions are cooking and after grill is lit, combine ground beef, Worcestershire sauce, and salt and pepper to taste in a large bowl.
Handling the meat as little as possible to avoid compacting it, mix ingredients together well.
Divide the mixture into 8 equal portions, and form into thin, round patties to fit the bread slices.
Cut the cheese into 4 pieces a little smaller that the patties.
Place a piece of cheese on 4 of the patties, cover with the remaining patties, and press edges together to seal and encase the cheese.
When grill is ready, brush the grill rack or broiler rack with vegetable oil.
Place the patties on the grill rack or under the broiler and cook until browned, about 4 minutes.
With a wide spatula, turn the patties and cook until done to preference, about 4 minutes longer for medium rare.
During the last few minutes of cooking, place the bread slices on the outer edges of the grill to toast lightly or prepare in toaster.
Top 4 slices of the bread with the patties, pile on the onion, and cover.
with the remaining bread slices.